The Scotland star joined the Stamford Bridge side in 2017 for a “significant fee”.
It is understood the deal was worth £500,000 with further add-ons and clauses inserted depending on how well Gilmour progressed.
The 20-year-old has made 22 appearances for the first-team and was part of the squad which won the Champions League in May.
There was speculation that Gilmour could return to Ibrox on loan for the season as he seeks regular first-team football. However, he is on the verge of joining Norwich City.
A move to the Premier League new boys for the season is expected to be completed imminently with the player set to complete a medical following his isolation after testing positive for Covid-19 on Scotland duty.